Solve 2(y-3x)^2+2(y+2x)(y-2x)-9x^2-2xy-(2y-x)^2

$$2 ( y - 3 x ) ^ { 2 } + 2 ( y + 2 x ) ( y - 2 x ) - 9 x ^ { 2 } - 2 x y - ( 2 y - x ) ^ { 2 }$$

Evaluate

$$-10xy$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-3x\right)^{2}.

$$2\left(y^{2}-6yx+9x^{2}\right)+2\left(y+2x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2 by y^{2}-6yx+9x^{2}.

$$2y^{2}-12yx+18x^{2}+2\left(y+2x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2 by y+2x.

$$2y^{2}-12yx+18x^{2}+\left(2y+4x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2y+4x by y-2x and combine like terms.

$$2y^{2}-12yx+18x^{2}+2y^{2}-8x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 2y^{2} and 2y^{2} to get 4y^{2}.

$$4y^{2}-12yx+18x^{2}-8x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 18x^{2} and -8x^{2} to get 10x^{2}.

$$4y^{2}-12yx+10x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$4y^{2}-12yx+x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine -12yx and -2xy to get -14yx.

$$4y^{2}-14yx+x^{2}-\left(2y-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2y-x\right)^{2}.

$$4y^{2}-14yx+x^{2}-\left(4y^{2}-4yx+x^{2}\right)$$

To find the opposite of 4y^{2}-4yx+x^{2}, find the opposite of each term.

$$4y^{2}-14yx+x^{2}-4y^{2}+4yx-x^{2}$$

Combine 4y^{2} and -4y^{2} to get 0.

$$-14yx+x^{2}+4yx-x^{2}$$

Combine -14yx and 4yx to get -10yx.

$$-10yx+x^{2}-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-10yx$$

Expand

$$-10xy$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-3x\right)^{2}.

$$2\left(y^{2}-6yx+9x^{2}\right)+2\left(y+2x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2 by y^{2}-6yx+9x^{2}.

$$2y^{2}-12yx+18x^{2}+2\left(y+2x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2 by y+2x.

$$2y^{2}-12yx+18x^{2}+\left(2y+4x\right)\left(y-2x\right)-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Use the distributive property to multiply 2y+4x by y-2x and combine like terms.

$$2y^{2}-12yx+18x^{2}+2y^{2}-8x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 2y^{2} and 2y^{2} to get 4y^{2}.

$$4y^{2}-12yx+18x^{2}-8x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 18x^{2} and -8x^{2} to get 10x^{2}.

$$4y^{2}-12yx+10x^{2}-9x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine 10x^{2} and -9x^{2} to get x^{2}.

$$4y^{2}-12yx+x^{2}-2xy-\left(2y-x\right)^{2}$$

Combine -12yx and -2xy to get -14yx.

$$4y^{2}-14yx+x^{2}-\left(2y-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2y-x\right)^{2}.

$$4y^{2}-14yx+x^{2}-\left(4y^{2}-4yx+x^{2}\right)$$

To find the opposite of 4y^{2}-4yx+x^{2}, find the opposite of each term.

$$4y^{2}-14yx+x^{2}-4y^{2}+4yx-x^{2}$$

Combine 4y^{2} and -4y^{2} to get 0.

$$-14yx+x^{2}+4yx-x^{2}$$

Combine -14yx and 4yx to get -10yx.

$$-10yx+x^{2}-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-10yx$$